Find the derivative of LaTeX: \displaystyle y = \frac{\left(- 8 x - 3\right)^{7} \left(x - 8\right)^{8} e^{x}}{\left(x + 6\right)^{7} \sqrt{8 x + 8} \left(9 x - 8\right)^{6}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{\left(- 8 x - 3\right)^{7} \left(x - 8\right)^{8} e^{x}}{\left(x + 6\right)^{7} \sqrt{8 x + 8} \left(9 x - 8\right)^{6}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 7 \ln{\left(- 8 x - 3 \right)} + 8 \ln{\left(x - 8 \right)}- 7 \ln{\left(x + 6 \right)} - \frac{\ln{\left(8 x + 8 \right)}}{2} - 6 \ln{\left(9 x - 8 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = 1 - \frac{54}{9 x - 8} - \frac{4}{8 x + 8} - \frac{7}{x + 6} + \frac{8}{x - 8} - \frac{56}{- 8 x - 3} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(1 - \frac{54}{9 x - 8} - \frac{4}{8 x + 8} - \frac{7}{x + 6} + \frac{8}{x - 8} - \frac{56}{- 8 x - 3}\right)\left(\frac{\left(- 8 x - 3\right)^{7} \left(x - 8\right)^{8} e^{x}}{\left(x + 6\right)^{7} \sqrt{8 x + 8} \left(9 x - 8\right)^{6}} \right)